Labeling something hate speech and trying to outlaw it is not an appropriate response to someone making claims. All it does is promote treating one group better than other groups which in turn breeds resentment.
If someone is spouting off lies and we can say they are lying beyond all doubt, then surely there is no need to silence them for their claims will easily be proven false and Noone of average (or even below average) intelligence will believe them. If, however, one or more parties have been lying or grossly exaggerating certain claims than I can understand wanting to silence anyone exposing such "alternate facts".
What's next? Labeling speech which is "clearly and beyond all doubt scientifically easily proven false as anti-scientific and Outlawing that as well?